Controllability

The state of a deterministic system, which is the set of values of all the system's state variables (those variables characterized by dynamic equations), completely describes the system at any given time. In particular, no information on the past of a system is needed to help in predicting the future, if the states at the present time are known and all current and future values of the control variables (those whose values can be chosen) are known.

Complete state controllability (or simply controllability if no other context is given) describes the ability of an external input (the vector of control variables) to move the internal state of a system from any initial state to any final state in a finite time interval.[1]^: 737

That is, we can informally define controllability as follows: If for any initial state ${\displaystyle \mathbf {x_{0}} }$ and any final state ${\displaystyle \mathbf {x_{f}} }$ there exists an input sequence to transfer the system state from ${\displaystyle \mathbf {x_{0}} }$ to ${\displaystyle \mathbf {x_{f}} }$ in a finite time interval, then the system modeled by the state-space representation is controllable. For the simplest example of a continuous, LTI system, the row dimension of the state space expression ${\displaystyle {\dot {\mathbf {x} }}=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t)}$ determines the interval; each row contributes a vector in the state space of the system. If there are not enough such vectors to span the state space of ${\displaystyle \mathbf {x} }$, then the system cannot achieve controllability. It may be necessary to modify ${\displaystyle \mathbf {A} }$ and ${\displaystyle \mathbf {B} }$ to better approximate the underlying differential relationships it estimates to achieve controllability.

Controllability does not mean that a reached state can be maintained, merely that any state can be reached.

Controllability does not mean that arbitrary paths can be made through state space, only that there exists a path within the prescribed finite time interval.

Consider the continuous linear system [note 1]

${\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)+B(t)\mathbf {u} (t)}$

${\displaystyle \mathbf {y} (t)=C(t)\mathbf {x} (t)+D(t)\mathbf {u} (t).}$

There exists a control ${\displaystyle u}$ from state ${\displaystyle x_{0}}$ at time ${\displaystyle t_{0}}$ to state ${\displaystyle x_{1}}$ at time ${\displaystyle t_{1}>t_{0}}$ if and only if ${\displaystyle x_{1}-\phi (t_{0},t_{1})x_{0}}$ is in the column space of

${\displaystyle W(t_{0},t_{1})=\int _{t_{0}}^{t_{1}}\phi (t_{0},t)B(t)B(t)^{T}\phi (t_{0},t)^{T}dt}$

where ${\displaystyle \phi }$ is the state-transition matrix, and ${\displaystyle W(t_{0},t_{1})}$ is the Controllability Gramian.

In fact, if ${\displaystyle \eta _{0}}$ is a solution to ${\displaystyle W(t_{0},t_{1})\eta =x_{1}-\phi (t_{0},t_{1})x_{0}}$ then a control given by ${\displaystyle u(t)=-B(t)^{T}\phi (t_{0},t)^{T}\eta _{0}}$ would make the desired transfer.

Note that the matrix ${\displaystyle W}$ defined as above has the following properties:

• ${\displaystyle W(t_{0},t_{1})}$ is symmetric
• ${\displaystyle W(t_{0},t_{1})}$ is positive semidefinite for ${\displaystyle t_{1}\geq t_{0}}$
• ${\displaystyle W(t_{0},t_{1})}$ satisfies the linear matrix differential equation

${\displaystyle {\frac {d}{dt}}W(t,t_{1})=A(t)W(t,t_{1})+W(t,t_{1})A(t)^{T}-B(t)B(t)^{T},\;W(t_{1},t_{1})=0}$

• ${\displaystyle W(t_{0},t_{1})}$ satisfies the equation

${\displaystyle W(t_{0},t_{1})=W(t_{0},t)+\phi (t_{0},t)W(t,t_{1})\phi (t_{0},t)^{T}}$[2]

The Controllability Gramian involves integration of the state-transition matrix of a system. A simpler condition for controllability is a rank condition analogous to the Kalman rank condition for time-invariant systems.

Consider a continuous-time linear system ${\displaystyle \Sigma }$ smoothly varying in an interval ${\displaystyle [t_{0},t]}$ of ${\displaystyle \mathbb {R} }$:

${\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)+B(t)\mathbf {u} (t)}$

${\displaystyle \mathbf {y} (t)=C(t)\mathbf {x} (t)+D(t)\mathbf {u} (t).}$

The state-transition matrix ${\displaystyle \phi }$ is also smooth. Introduce the n x m matrix-valued function ${\displaystyle M_{0}(t)=\phi (t_{0},t)B(t)}$ and define

${\displaystyle M_{k}(t)}$ = ${\displaystyle {\frac {\mathrm {d^{k}} M_{0}}{\mathrm {d} t^{k}}}(t),k\geqslant 1}$.

Consider the matrix of matrix-valued functions obtained by listing all the columns of the ${\displaystyle M_{i}}$, ${\displaystyle i=0,1,\ldots ,k}$:

${\displaystyle M^{(k)}(t):=\left[M_{0}(t),\ldots ,M_{k}(t)\right]}$.

If there exists a ${\displaystyle {\bar {t}}\in [t_{0},t]}$ and a nonnegative integer k such that ${\displaystyle \operatorname {rank} M^{(k)}({\bar {t}})=n}$, then ${\displaystyle \Sigma }$ is controllable.[3]

If ${\displaystyle \Sigma }$ is also analytically varying in an interval ${\displaystyle [t_{0},t]}$, then ${\displaystyle \Sigma }$ is controllable on every nontrivial subinterval of ${\displaystyle [t_{0},t]}$ if and only if there exists a ${\displaystyle {\bar {t}}\in [t_{0},t]}$ and a nonnegative integer k such that ${\displaystyle \operatorname {rank} M^{(k)}(t_{i})=n}$.[3]

The above methods can still be complex to check, since it involves the computation of the state-transition matrix ${\displaystyle \phi }$. Another equivalent condition is defined as follow. Let ${\displaystyle B_{0}(t)=B(t)}$, and for each ${\displaystyle i\geq 0}$, define

${\displaystyle B_{i+1}(t)}$= ${\displaystyle A(t)B_{i}(t)-{\frac {\mathrm {d} }{\mathrm {d} t}}B_{i}(t).}$

In this case, each ${\displaystyle B_{i}}$ is obtained directly from the data ${\displaystyle (A(t),B(t)).}$ The system is controllable if there exists a ${\displaystyle {\bar {t}}\in [t_{0},t]}$ and a nonnegative integer ${\displaystyle k}$ such that ${\displaystyle {\textrm {rank}}(\left[B_{0}({\bar {t}}),B_{1}({\bar {t}}),\ldots ,B_{k}({\bar {t}})\right])=n}$.[3]

For a discrete-time linear state-space system (i.e. time variable ${\displaystyle k\in \mathbb {Z} }$) the state equation is

${\displaystyle {\textbf {x}}(k+1)=A{\textbf {x}}(k)+B{\textbf {u}}(k)}$

where ${\displaystyle A}$ is an ${\displaystyle n\times n}$ matrix and ${\displaystyle B}$ is a ${\displaystyle n\times r}$ matrix (i.e. ${\displaystyle \mathbf {u} }$ is ${\displaystyle r}$ inputs collected in a ${\displaystyle r\times 1}$ vector). The test for controllability is that the ${\displaystyle n\times nr}$ matrix

${\displaystyle {\mathcal {C}}={\begin{bmatrix}B&AB&A^{2}B&\cdots &A^{n-1}B\end{bmatrix}}}$

has full row rank (i.e., ${\displaystyle \operatorname {rank} ({\mathcal {C}})=n}$). That is, if the system is controllable, ${\displaystyle {\mathcal {C}}}$ will have ${\displaystyle n}$ columns that are linearly independent; if ${\displaystyle n}$ columns of ${\displaystyle {\mathcal {C}}}$ are linearly independent, each of the ${\displaystyle n}$ states is reachable by giving the system proper inputs through the variable ${\displaystyle u(k)}$.

Nonlinear systems in the control-affine form

${\displaystyle {\dot {\mathbf {x} }}=\mathbf {f(x)} +\sum _{i=1}^{m}\mathbf {g} _{i}(\mathbf {x} )u_{i}}$

are locally accessible about ${\displaystyle x_{0}}$ if the accessibility distribution ${\displaystyle R}$ spans ${\displaystyle n}$ space, when ${\displaystyle n}$ equals the rank of ${\displaystyle x}$ and R is given by:[4]

${\displaystyle R={\begin{bmatrix}\mathbf {g} _{1}&\cdots &\mathbf {g} _{m}&[\mathrm {ad} _{\mathbf {g} _{i}}^{k}\mathbf {\mathbf {g} _{j}} ]&\cdots &[\mathrm {ad} _{\mathbf {f} }^{k}\mathbf {\mathbf {g} _{i}} ]\end{bmatrix}}.}$

Here, ${\displaystyle [\mathrm {ad} _{\mathbf {f} }^{k}\mathbf {\mathbf {g} } ]}$ is the repeated Lie bracket operation defined by

${\displaystyle [\mathrm {ad} _{\mathbf {f} }^{k}\mathbf {\mathbf {g} } ]={\begin{bmatrix}\mathbf {f} &\cdots &j&\cdots &\mathbf {[\mathbf {f} ,\mathbf {g} ]} \end{bmatrix}}.}$

The controllability matrix for linear systems in the previous section can in fact be derived from this equation.

If a discrete control system is null-controllable, it means that there exists a controllable ${\displaystyle u(k)}$ so that ${\displaystyle x(k_{0})=0}$ for some initial state ${\displaystyle x(0)=x_{0}}$. In other words, it is equivalent to the condition that there exists a matrix ${\displaystyle F}$ such that ${\displaystyle A+BF}$ is nilpotent.

This can be easily shown by controllable-uncontrollable decomposition.

Output controllability is the related notion for the output of the system (denoted y in the previous equations); the output controllability describes the ability of an external input to move the output from any initial condition to any final condition in a finite time interval. It is not necessary that there is any relationship between state controllability and output controllability. In particular:

• A controllable system is not necessarily output controllable. For example, if matrix D = 0 and matrix C does not have full row rank, then some positions of the output are masked by the limiting structure of the output matrix, and therefore unachievable. Moreover, even though the system can be moved to any state in finite time, there may be some outputs that are inaccessible by all states. A trivial numerical example uses D=0 and a C matrix with at least one row of zeros; thus, the system is not able to produce a non-zero output along that dimension.
• An output controllable system is not necessarily state controllable. For example, if the dimension of the state space is greater than the dimension of the output, then there will be a set of possible state configurations for each individual output. That is, the system can have significant zero dynamics, which are trajectories of the system that are not observable from the output. Consequently, being able to drive an output to a particular position in finite time says nothing about the state configuration of the system.

For a linear continuous-time system, like the example above, described by matrices ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$, and ${\displaystyle D}$, the ${\displaystyle m\times (n+1)r}$ output controllability matrix

${\displaystyle {\begin{bmatrix}CB&CAB&CA^{2}B&\cdots &CA^{n-1}B&D\end{bmatrix}}}$

has full row rank (i.e. rank ${\displaystyle m}$) if and only if the system is output controllable.[1]^: 742

In systems with limited control authority, it is often no longer possible to move any initial state to any final state inside the controllable subspace. This phenomenon is caused by constraints on the input that could be inherent to the system (e.g. due to saturating actuator) or imposed on the system for other reasons (e.g. due to safety-related concerns). The controllability of systems with input and state constraints is studied in the context of reachability[5] and viability theory.[6]

In the so-called behavioral system theoretic approach due to Willems (see people in systems and control), models considered do not directly define an input–output structure. In this framework systems are described by admissible trajectories of a collection of variables, some of which might be interpreted as inputs or outputs.

A system is then defined to be controllable in this setting, if any past part of a behavior (trajectory of the external variables) can be concatenated with any future trajectory of the behavior in such a way that the concatenation is contained in the behavior, i.e. is part of the admissible system behavior.[7]^: 151

A slightly weaker notion than controllability is that of stabilizability. A system is said to be stabilizable when all uncontrollable state variables can be made to have stable dynamics. Thus, even though some of the state variables cannot be controlled (as determined by the controllability test above) all the state variables will still remain bounded during the system's behavior.[8]

Let T ∈ Т and x ∈ X (where X is the set of all possible states and Т is an interval of time). The reachable set from x in time T is defined as:[3]

${\displaystyle R^{T}{(x)}=\left\{z\in X:x{\overset {T}{\rightarrow }}z\right\}}$, where xT→z denotes that there exists a state transition from x to z in time T.

For autonomous systems the reachable set is given by :

${\displaystyle \mathrm {Im} (R)=\mathrm {Im} (B)+\mathrm {Im} (AB)+....+\mathrm {Im} (A^{n-1}B)}$,

where R is the controllability matrix.

In terms of the reachable set, the system is controllable if and only if ${\displaystyle \mathrm {Im} (R)=\mathbb {R} ^{n}}$.

Proof We have the following equalities:

${\displaystyle R=[B\ AB....A^{n-1}B]}$

${\displaystyle \mathrm {Im} (R)=\mathrm {Im} ([B\ AB....A^{n-1}B])}$

${\displaystyle \mathrm {dim(Im} (R))=\mathrm {rank} (R)}$

Considering that the system is controllable, the columns of R should be linearly independent. So:

${\displaystyle \mathrm {dim(Im} (R))=n}$

${\displaystyle \mathrm {rank} (R)=n}$

${\displaystyle \mathrm {Im} (R)=\mathbb {R} ^{n}\quad \blacksquare }$

A related set to the reachable set is the controllable set, defined by:

${\displaystyle C^{T}{(x)}=\left\{z\in X:z{\overset {T}{\rightarrow }}x\right\}}$.

The relation between reachability and controllability is presented by Sontag:[3]

(a) An n-dimensional discrete linear system is controllable if and only if:

${\displaystyle R(0)=R^{k}{(0)=X}}$ (Where X is the set of all possible values or states of x and k is the time step).

(b) A continuous-time linear system is controllable if and only if:

${\displaystyle R(0)=R^{e}{(0)=X}}$ for all e>0.

if and only if ${\displaystyle C(0)=C^{e}{(0)=X}}$ for all e>0.

Example Let the system be an n dimensional discrete-time-invariant system from the formula:

Φ(n,0,0,w)=${\displaystyle \sum \limits _{i=1}^{n}A^{i-1}Bw(n-1)}$ (Where Φ(final time, initial time, state variable, restrictions) is defined is the transition matrix of a state variable x from a initial time 0 to a final time n with some restrictions w).

It follows that the future state is in ${\displaystyle R^{k}{(0)}}$ ⇔ it is in the image of the linear map:

Im(R)=R(A,B)≜ Im(${\displaystyle [B\ AB....A^{n-1}B]}$),

which maps,

${\displaystyle u^{n}}$→X

When ${\displaystyle u=K^{m}}$ and ${\displaystyle X=K^{n}}$ we identify R(A,B) with a n by nm matrix whose columns are the columns of ${\displaystyle B,\ AB,....,A^{n-1}B}$ in that order. If the system is controllable the rank of ${\displaystyle [B\ AB....A^{n-1}B]}$ is n. If this is truth, the image of the linear map R is all of X. Based on that, we have:

${\displaystyle R(0)=R^{k}{(0)=X}}$ with XЄ${\displaystyle \mathbb {R} ^{n}}$.

• Observability
• Hautus lemma

• MATLAB function for checking controllability of a system
• Mathematica function for checking controllability of a system